= Solution
Let $b$ denote <buoyancy> in a stably stratified <Boussinesq approximation>. <Geostrophic balance> and <hydrostatic balance> imply the <thermal wind> relation $f_0U_z=-\bar b_y$. Consequently the specified vertical shear requires
$$
\boxed{\bar b_y=-f_0\Lambda,\qquad \bar b_z=N^2>0}.
$$
For northern-hemisphere $f_0>0$ and $\Lambda>0$, the mean <potential temperature> decreases northward and increases upward: $\bar\Theta_y=-(\Theta_0/g)f_0\Lambda$, $\bar\Theta_z=(\Theta_0/g)N^2$. The corresponding mean isentropes slope upward toward the north, with $dz/dy=f_0\Lambda/N^2$. Stable potential-temperature stratification does not require the actual <temperature> to increase upward; the dry adiabatic lapse also enters that comparison.
In the <rigid-lid constant-shear baroclinic model>, perturbation <quasi-geostrophic potential vorticity> is
$$
q'=\nabla_h^2\psi'+\frac{f_0^2}{N^2}\psi'_{zz}.
$$
The mean interior potential-vorticity gradient vanishes on this constant-$f_0$, constant-$N$, linear-shear basic state. Linear <advection> of that perturbation therefore becomes $(\partial_t+\Lambda z\partial_x)q'=0$, the supplied interior equation. It retains slow balanced motion and omits <friction>, planetary-vorticity gradients and nonlinear disturbance <advection>.
Substitution of $\psi'=\Psi(z)\cos(ly)e^{ik(x-ct)}$ gives
$$
ik(\Lambda z-c)\left[\frac{f_0^2}{N^2}\Psi''-(k^2+l^2)\Psi\right]=0.
$$
A possible branch has zero interior perturbation <potential vorticity>, so
$$
\boxed{\Psi''-K^2\Psi=0,\qquad
\Psi=A\cosh(Kz)+B\sinh(Kz),\qquad K=\frac{N\sqrt{k^2+l^2}}{|f_0|}}.
$$
Both supplied applications have $f_0>0$, so this equals the source's expression. For an unstable mode the nonzero <imaginary part> of the <phase speed> prevents a real critical level $\Lambda z=c$; for real speeds the displayed solution remains a valid zero-interior-PV branch away from, and smoothly through, any critical level. Taking the <real part> gives the physical disturbance.
The rigid lids enforce $w'=0$. Linearized adiabatic <buoyancy> conservation is
$$
(\partial_t+\Lambda z\partial_x)b'+v'\bar b_y+N^2w'=0.
$$
Use the geostrophic conventions $v'=\psi'_x$ and $b'=f_0\psi'_z$. At a lid this gives $(\partial_t+\Lambda z\partial_x)\psi'_z-\Lambda\psi'_x=0$, hence
$$
\boxed{-c\Psi'(0)-\Lambda\Psi(0)=0,\qquad
(\Lambda H-c)\Psi'(H)-\Lambda\Psi(H)=0}.
$$
These are the no-normal-flow condition expressed through boundary <buoyancy> <advection>, not conditions of zero horizontal velocity. Boundary <temperature> anomalies can interact even though this solution has no interior perturbation PV, allowing <baroclinic instability> to extract mean <available potential energy>.
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